A computer-implemented method and system for implementing a n-fold fermionic excitation generator using linear combination of directly differentiable operators on a quantum computer. Computer-readable data is generated and stored which when executed on the quantum computer, causes a quantum circuit of the quantum computer to execute repeatedly to perform a sequence of operations that implements the unitary (I) generated by a fermionic n-fold excitation operator G. The Gradient with respect to the angle Θ of arbitrary expectation values involving the unitary operation can, in the general case, be evaluated by four expectation values obtained from replacing the corresponding unitary with fermionic shift operations (II). Fermionic shift operations can be constructed through the original unitary and unitary operations generated by the nullspace projector P0 of the fermionic excitation generator. Other operators and generators are disclosed.